There is a very simple way to understand this. Forget about the possessive aspect and ask yourself the question:
Can 'each other' ever be plural? The answer is no. Here's an example:
Imagine a football team. Every member gives a Christmas card to every other member. There are many people and many cards. Does that mean 'other' should be plural? Let's try:
The members of the team all gave each others a Christmas card.
Any native speaker, whether they have a knowledge of grammar or not, will say that that is wrong. They will tell you to say:
The members of the team all gave each other a Christmas card.
The reason is that 'each' is always singular and can only modify a singular noun.
Another way to see the same thing is to consider 'each other' as meaning 'each other person'. You would say each other person's belongings but never each other persons' belongings.
Answer
It is always each other and there is no context in which it could be each others. This holds true even if you add the apostrophe to make the expression possessive.
The correct usage would would be:
The members of the team like their cards and admired each other's presents.
Update in response to a comment
The following passage supports my argument that 'each other' is correct and 'each others' is incorrect.
Every, like each, is always used with a singular noun form and therefore with a singular verb form in English because we are counting the things or people that we are talking about separately one by one:
Every child in the class plays a musical instrument. Every instrument belonging to the school is tuned regularly. everybody / everyone / everything
